Notation and Modeling Addition of Integers

Learning Outcomes

Model addition of integers with color counters

Now that we have located positive and negative numbers on the number line, it is time to discuss arithmetic operations with integers. Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more difficult. This difficulty relates to the way the brain learns. The brain learns best by working with objects in the real world and then generalizing to abstract concepts. Toddlers learn quickly that if they have two cookies and their older brother steals one, they have only one left. This is a concrete example of

2 - 1

. Children learn their basic addition and subtraction facts from experiences in their everyday lives. Eventually, they know the number facts without relying on cookies. Addition and subtraction of negative numbers have fewer real world examples that are meaningful to us. Math teachers have several different approaches, such as number lines, banking, temperatures, and so on, to make these concepts real. We will model addition and subtraction of negatives with two color counters. We let a blue counter represent a positive and a red counter will represent a negative. This figure has a blue circle labeled positive and a red circle labeled negative.

This figure has a blue circle labeled positive and a red circle labeled negative.

If we have one positive and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero as summarized in the figure below. A blue counter represents

+1

. A red counter represents

-1

. Together they add to zero. This figure has a blue circle over a red circle. Beside them is the statement 1 plus negative 1 equals 0.

Doing the Manipulative Mathematics activity "Addition of signed Numbers" will help you develop a better understanding of adding integers. We will model four addition facts using the numbers

5,-5\text{ and }3,-3

5+3 - 5+\left(-3\right)-5+35+\left(-3\right)

example

Model:

5+3

. Solution:

Interpret the expression.	$5+3$ means the sum of $5$ and $3$ .
Model the first number. Start with $5$ positives.
Model the second number. Add $3$ positives.
Count the total number of counters.
The sum of $5$ and $3$ is $8$ .	$5+3=8$

try it

Model the expression.

2+4

Answer: This figure has six pink circles in a row, representing positive counters. The first two circles are separated from the following four circles. $6$

Model the expression.

2+5

Answer: This figure has seven pink circles in a row, representing positive counters. Two circles are separated from the following 5 circles. $7$

example

Model:

-5+\left(-3\right)

Answer: Solution:

Interpret the expression.	$-5+\left(-3\right)$ means the sum of $-5$ and $-3$ .
Model the first number. Start with $5$ negatives.
Model the second number. Add $3$ negatives.
Count the total number of counters.
The sum of $−5$ and $−3$ is $−8$ .	$-5+-3=-8$

try it

Model the expression.

-2+\left(-4\right)

Answer: This figure shows a row of 6 dark pink circles, representing negative counters. They are grouped by 2 circles followed by 4 circles. $−6$

Model the expression.

-2+\left(-5\right)

Answer: This figure shows a row of 7 dark pink circles, representing negative counters. They are grouped by 2 circles followed by 5 circles. $−7$

The first and second examples are very similar. The first example adds

5

positives and

3

positives—both positives. The second example adds

5

negatives and

3

negatives—both negatives. In each case, we got a result of

\text{8-either}8

positives or

8

negatives. When the signs are the same, the counters are all the same color. Now let’s see what happens when the signs are different.

example

Model:

-5+3

Answer: Solution:

Interpret the expression.	$-5+3$ means the sum of $-5$ and $3$ .
Model the first number. Start with $5$ negatives.
Model the second number. Add $3$ positives.
Remove any neutral pairs.
Count the result.
The sum of $−5$ and $3$ is $−2$ .	$-5+3=-2$

Notice that there were more negatives than positives, so the result is negative.

try it

Model the expression, and then simplify:

2+\left(-4\right)

Answer: This figure shows two rows of counter circles. The first row has 2 dark pink circles, representing negative counters. The second row has 4 light pink circles, representing positive counters. $−2$

Model the expression, and then simplify:

2+\left(-5\right)

Answer: This figure shows two rows of counter circles. The first row has 2 light pink circles, representing positive counters. The second row has 5 dark pink circles, representing negative counters. $−3$

example

Model:

5+\left(-3\right)

Answer: Solution:

Interpret the expression.	$5+\left(-3\right)$ means the sum of $5$ and $-3$ .
Model the first number. Start with $5$ positives.
Model the second number. Add $3$ negatives.
Remove any neutral pairs.
Count the result.
The sum of $5$ and $−3$ is $2$ .	$5+\left(-3\right)=2$

try it

Model the expression, and then simplify:

\left(-2\right)+4

Answer: This figure shows two rows of counter circles. The first row has 2 light pink circles, representing positive counters. The second row has 4 dark pink circles, representing negative counters. $−2$

Model the expression:

\left(-2\right)+5

Answer: This figure shows two rows of counters. The first row shows 2 dark pink circles, representing negative counters. The second row has 5 light pink circles, representing negative counters. The neutral pairs of one positive and one negative counter are circled leaving three positive counters. $3$